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14 CAUCHY SEQUENCES, COMPLETE SPACES 53

(3.14.2) If (x_n) is a Cauchy sequence, any cluster value of (x_n) is a limit
of(x_n\

Indeed, if b is a cluster value of (x_n), given e > 0, there is n₀ such that
p^n₀ and q^n₀ imply d(x_p, x_q) < e/2; on the other hand, by (3.13.11)
there is a p_Q ^ n_Q such that d(b, x_po) < e/2; by the triangle inequality, it
follows that d(b, x_n) ^ e for any n ^ /7₀.

A metric space E is called complete if any Cauchy sequence in E is con-
vergent (to a point of E, of course).

(3.14.3) The real line R is a complete metric space.

Let (x_n) be a Cauchy sequence of real numbers. Define the sequence
(n_k) of integers by induction in the following way: /?₀ = 1 and 77^ + 1 is the
smallest integer > n_k such that, forp^ n_k+l and q ^ n_k+l9 \x_p — x_q\ < l/2^k+2;
the possibility of the definition follows from the fact that (x_n) is a Cauchy
sequence. Let l_k be the closed interval [x_nk— 2~^k, x_nk 4- 2~~^k]; we have
Ift + i ^c I_A, for \x_ttk - x»_k ₊ _{\ < 2"*™¹; on the other hand, for m ^ n_k, x_m e l_kby definition. Now from axiom (IV) (Section 2.1) it follows that the
nested intervals 1^ have a nonempty intersection; let a be in 1^ for all k. Then
it is clear that a — x ^.2~^k ⁺ ^l for all m^n, hence a = lim x.

(3,14.4) If a subspace Fofa metric space E is complete, F is closed in E.

Indeed, any point a e F is the limit of a sequence (x_n) of points of F by
(3.13.13). The sequence (x_n) is a Cauchy sequence by (3.14.1), hence by
assumption converges to a point b in F; but by (3.13.3) b = a, hence a e F;
this shows F = F. Q.E.D.

(3.14.5) In a complete metric space E, any closed subset F is a complete
subspace.

For a Cauchy sequence (x_n) of points of F converges by assumption to
a point a e E, and as the x_n belong to F, a e F = F by (3.13.7).

Theorems (3.14.4) and (3.14.5) immediately enable one to give examples
both of complete and of noncomplete spaces, starting from the fact that the
real line is complete.



	14 CAUCHY SEQUENCES, COMPLETE SPACES 53 (3.14.2) If (x_n) is a Cauchy sequence, any cluster value of (x_n) is a limit of(x_n\ Indeed, if b is a cluster value of (x_n), given e > 0, there is n₀ such that p^n₀ and q^n₀ imply d(x_p, x_q) < e/2; on the other hand, by (3.13.11) there is a p_Q ^ n_Q such that d(b, x_po) < e/2; by the triangle inequality, it follows that d(b, x_n) ^ e for any n ^ /7₀. A metric space E is called complete if any Cauchy sequence in E is con- vergent (to a point of E, of course).

	(3.14.3) The real line R is a complete metric space. Let (x_n) be a Cauchy sequence of real numbers. Define the sequence (n_k) of integers by induction in the following way: /?₀ = 1 and 77^ + 1 is the smallest integer > n_k such that, forp^ n_k+l and q ^ n_k+l9 \x_p — x_q\ < l/2^k+2; the possibility of the definition follows from the fact that (x_n) is a Cauchy sequence. Let l_k be the closed interval [x_nk— 2~^k, x_nk 4- 2~~^k]; we have Ift + i ^c I_A, for \x_ttk - x»_k ₊ _{\ < 2"™¹; on the other hand, for m ^ n_k, x_m e l_kby definition. Now from axiom (IV) (Section 2.1) it follows that the nested intervals 1^ have a nonempty intersection; let a be in 1^ for all k. Then it is clear that a — x ^.2~^k ⁺ ^l* for all m^n, hence a = lim x.

	(3,14.4) If a subspace Fofa metric space E is complete, F is closed in E. Indeed, any point a e F is the limit of a sequence (x_n) of points of F by (3.13.13). The sequence (x_n) is a Cauchy sequence by (3.14.1), hence by assumption converges to a point b in F; but by (3.13.3) b = a, hence a e F; this shows F = F. Q.E.D.

	(3.14.5) In a complete metric space E, any closed subset F is a complete subspace. For a Cauchy sequence (x_n) of points of F converges by assumption to a point a e E, and as the x_n belong to F, a e F = F by (3.13.7). Theorems (3.14.4) and (3.14.5) immediately enable one to give examples both of complete and of noncomplete spaces, starting from the fact that the real line is complete.